Stringent emission regulations and improved fuel economy and performance dictate more advanced powertrain configurations than standard port-fuel injected gasoline engines. Modern state-of-the-art powertrain systems may combine several power sources (internal combustion engines, electric motors, fuel cells, etc.) and various exhaust after-treatment devices (catalytic converters including lean NOx traps, particulate filters, etc.) in addition to conventional engine subsystems such as turbochargers and exhaust gas recirculation. The determination of the way in which these systems need be operated to meet driver's torque demand, performance and fuel economy expectations while satisfying federal emission regulations is a complex and a multi-objective optimal control problem.
The advanced powertrains are characterized by an increased number of control inputs as compared to the conventional powertrains and by significant subsystem interactions. As a result, optimizing the operation of these powertrains is significantly more difficult than optimizing a conventional powertrain. Additional complications arise due to the need to treat engine subsystems with energy or emission storage mechanisms, e.g., a battery or an emission trap. These storage mechanisms change the nature of the optimization task in a fundamental way.
Optimization procedures for the conventional gasoline or diesel powertrains are well known (e.g., Rao, H. S., Cohen, A. I., Tennant, J. A., and Voorhies, K. L., "Engine Control Optimization Via Nonlinear Programming", SAE Paper No. 790177; Rishavy, E. A., Hamilton, S. C., Ayers, J. A., Keane, M. A., "Engine Control Optimization for Best Fuel Economy with Emission Constraints", SAE Paper No. 770075; and Scmitz, G., Oligschlager, U., and Eifler, G., "Automated System for Optimized Calibration of Engine Management Systems", SAE Paper No. 940151). These references make extensive use of the "quasi-static" assumption that substantially simplifies the optimization problem. This "quasi-static" assumption is that the internal combustion engine fuel consumption and feedgas emission values at any given time instant are static functions of engine speed, engine torque and control variables (such as fueling rate, spark timing, EGR valve position, etc.) at the same time instant. Steady-state engine mapping data generated experimentally or from a high fidelity simulation model are typically used to develop these static functions. This "quasi-static" assumption is appropriate for establishing trends and relative effects for the warmed-up operation. Once the optimized strategies have been generated under the "quasi-static" assumption, the actual numbers for emissions and fuel economy are typically validated, either experimentally or on a more detailed powertrain simulation model that incorporates transient effects. This "quasi-static" assumption is also used as a basis for several simulation and modeling packages.
To illustrate the use of the "quasi-static" assumption, consider, for example, the optimization procedure for a conventional port-fuel injected (PFI) spark ignition engine equipped with a Three-Way-Catalyst (TWC). First the engine speed and engine torque trajectories are derived from a vehicle speed profile (FIG. 1) using the estimates of the vehicle mass, tire radius, aerodynamic drag coefficient, frontal area, rolling resistance coefficient, gear ratios, shift schedule, idle speed value and power losses in the drivetrain. The time trajectory of the engine speed (N.sub.e) and the engine torque (.tau..sub.e) is then a prescribed two dimensional vector w(t)=[N.sub.e (t).tau..sub.e (t)].sup.T. Next, a discrete grid on the engine speed and engine torque values is introduced which divides the engine speed/engine torque plane into M rectangular cells, C.sup.i, i=1, . . . , M. Let T(i) be the total time the engine operates in the cell C.sup.i over the specified drive cycle while w.sup.i is the speed/torque vector corresponding to the center of the cell C.sup.i. For each of the cells C.sup.i, the values of the control inputs u.sup.i (fueling rate, spark timing, EGR rate, etc.) must be prescribed so that the fuel consumption over the drive-cycle is minimized while the emission constraints are met: ##EQU1##
subject to ##EQU2##
Here W.sub.f (u,w) denotes the fueling rate in gram per second, W.sub.s.sub..sub.j (u,w) is the mass flow rate of the jth regulated emission species (oxides of nitrogen (NO.sub.x), carbon monoxide (CO) and hydrocarbons (HC)) out of the tailpipe in gram per second and g.sub.j is the emission limit for the jth species, j=1,2,3. The representation of the objective function and constraints as a sum of independent terms (separability property) has been made possible by the "quasi-static" assumption on the engine operation and a similar assumption on "quasi-static" behavior of the TWC conversion efficiencies (valid for engine operation around stoichiometry). The application of the Lagrange Duality reduces the problem to a two-stage (inner loop/outer loop) optimization problem. In the first stage (inner loop), for each cell C.sup.i the cost function of the form ##EQU3##
is minimized with respect to u.sup.i, where .lambda..sub.j are the Lagrange multipliers. The same values of .lambda..sub.j are used for every cell. The optimization searches for a minimum either using regressions for W.sub.f and W.sub.s.sub..sub.j or directly on a finite set of experimental data points. Hence, a calibration is generated that prescribes the values of the control inputs u.sup.i =u.sup.i *(.lambda.) as functions of the Lagrange multipliers .lambda..sub.j, j=1,2,3, and the value of the dual function EQU .theta.(.lambda.)=.SIGMA..sub.i=1.sup.M T(i)F.sup.i (u.sup.i *(.lambda.),w.sup.i,.lambda.)
can be calculated. The outer loop of the optimization adjusts the Lagrange multipliers to achieve the desired objectives via the maximization of .theta.(.lambda.). The feasibility of the powertrain is established if the maximum of .theta. exists at some value of .lambda.=.lambda.*. Under appropriate additional assumptions the static calibration corresponding to .lambda.*, u.sup.i *(.lambda.*), provides the best emission constrained fuel economy over the specified drive cycle.